The main goal of this proposal is to reacha better mathematical understanding ofthe dynamics of quantum mechanicalsystems. In particular I plan to workon the following three projects alongthis direction. A. Effective EvolutionEquations for Macroscopic Systems.The derivation of effective evolutionequations from first principle microscopictheories is a fundamental task of statisticalmechanics. I have been involved inseveral projects related to the derivationof the Hartree and the Gross-Piteavskiiequation from many body quantumdynamics. I plan to continue to work onthese problems and to use these resultsto obtain new information on the manybody dynamics. B. Spectral Propertiesof Random Matrices. The correlationsamong eigenvalues of large randommatrices are expected to be independentof the distribution of the entries. Thisconjecture, known as universality, isof great importance for random matrixtheory. In collaboration with L. Erdos andH.-T. Yau, we established the validity ofWigner's semicircle law onmicroscopic scales, and we proved theemergence of eigenvalue repulsion. Inthe future, we plan to continue to studyWigner matrices to prove, on the longerterm, universality. C. Locality Estimates inQuantum Dynamics. Anharmonic latticesystems are very important models innon-equilibrium statistical mechanics.With B. Nachtergaele, H. Raz, and R.Sims, we proved Lieb-Robinson typeinequalities (giving an upper bound onthe speed of propagation of signals), fora certain class of anharmonicity. Next, weplan to extend these results to a largerclass of anharmonic potentials, and toapply these bounds to establish otherfundamental properties of the dynamicsof anharmonic systems, such as theexistence of its thermodynamical limit.